A public-key encryption method and system based on a novel LWE problem

Abstract

This invention belongs to the field of information security technology and discloses a public-key encryption method and system based on a novel LWE problem. The method includes defining a novel LWE problem based on a symmetric group over an n-ary set; constructing a public-key encryption scheme based on the novel LWE problem, wherein the public-key encryption scheme includes key generation by the decryptor, encryption by the encryptor, and decryption by the decryptor. This invention is based on a symmetric group over a non-commutative n-ary set, utilizes the corresponding group ring to construct the LWE problem, and proposes a public-key encryption scheme based on this novel LWE problem, avoiding the security impact of current analyses on ideal lattices.

Description

Technical Field

[0001] This invention relates to the field of information security technology, and in particular to a public-key encryption method and system based on a novel LWE problem. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Most public-key cryptography schemes currently used on the internet are based on integer factorization and the discrete logarithm problem, such as ElGamal and RSA. However, Shor presented an efficient quantum algorithm for solving these computationally difficult problems under classical computation. With the continuous progress in the development of quantum computers, these cryptographic schemes face a potential threat.

[0004] To address emerging threats, post-quantum cryptography has emerged, aiming to study the security of cryptographic algorithms in quantum environments. Many international associations and organizations (such as NIST, ISO, and ITU-T) have launched a series of quantum-resistant cryptography standardization projects. Lattice-based cryptography is one of the most attractive areas in post-quantum cryptography, and its comprehensive advantages, including worst-case to average-case difficulty reduction, high implementation efficiency, and support for multiple cryptographic functions, have led to its extensive research and application over the past decade. Lattice cryptography is poised to become the mainstream approach in post-quantum cryptography. Its unique advantages, including strong security, rapid implementation, and versatility in many applications, have garnered significant attention over the past decade.

[0005] To improve the efficiency of lattice-based schemes, researchers have employed special algebraic structures. The first lattice-based scheme to use an algebraic structure was the NTRU public-key cryptosystem, first proposed by Hoffstein, Pipher, and Silverman in 1996. It is used for rings of prime numbers. The cyclic properties of the ring provide efficiency for the NTRU scheme. Regev introduced the Learning with Errors (LWE) problem by generalizing the classical parity noise learning problem. He proved the hardness of LWE by providing quantum reductions from approximate SVP and SIVP problems to LWE under specific parameters. Later, Peikert showed a classical reduction from approximate SVP to LWE under stricter constraints. To improve efficiency, researchers proposed constructing a Ring Learning with Errors (RLWE) problem using algebraic integer rings. Furthermore, they demonstrated the reduction from a certain ideal lattice problem to the RLWE problem through different methods, thus establishing the hardness of RLWE. Cryptosystems based on RLWE are more efficient in terms of key size and encryption / decryption complexity. However, the security of these systems is based on conjectural puzzles on ideal lattices rather than on general lattices. However, there may be more efficient algorithms for solving the hard problem of ideal lattices on algebraic integer rings than for general lattice problems. For example, Cramer and Ducas et al. gave an approximation factor for solving the principal ideal lattice of the cyclotomic number field. This involves methods to approximate the SVP problem. Related attacks exploit properties such as the commutative structure of algebraic integer rings. Summary of the Invention

[0006] To address the aforementioned issues, this invention proposes a public-key encryption method and system based on a novel LWE problem. It utilizes a symmetric group on a non-commutative n-ary set as a foundation, constructs an LWE problem using the corresponding group ring, and proposes a public-key encryption scheme based on this novel LWE problem, thus avoiding the security impact of current analyses of ideal lattices.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] In a first aspect, the present invention provides a public-key encryption method based on a novel LWE problem, comprising the following steps:

[0009] A novel LWE problem is defined based on the symmetric group over n-ary sets;

[0010] A public-key encryption scheme is constructed based on a novel LWE problem. The public-key encryption scheme includes key generation by the decryptor, encryption of messages by the encryptor, and decryption of messages by the decryptor.

[0011] As an alternative implementation, a novel LWE problem is defined based on the symmetric group over an n-ary set, specifically:

[0012] Given a symmetric group S on an n-ary set n Construct group ring R q =Fq [S n ], where q is the modulus, F q It is a finite field modulo q, given a vector Let β be a real number and y be a vector y = xs + t, where m represents the dimension of the vector. U(·) is a uniform distribution. express Given a Gaussian distribution with upper deviation β, and based on symmetric groups and constructed group rings, find the secret s for x and y.

[0013] As an alternative implementation, the group ring is constructed based on a finite group and a commutative ring, where the symmetric group is the set of all permutations on an n-ary set.

[0014] As an alternative implementation method, the decryption key generation is as follows:

[0015] The decryptor selects a uniformly random element a∈R q And two random small elements s∈R, in, R represents q The output s is a Gaussian distribution with upper variance β, and is used as the private key. As a public key.

[0016] As an alternative implementation method, the encryptor encrypts the message, specifically as follows:

[0017] The cryptographer selects three random elements. Calculate the parameter u = r·a + e1modq and the parameter Output (u,v) as the ciphertext of the message to be encrypted.

[0018] As an alternative implementation method, the decryptor decrypts the message as follows:

[0019]

[0020] For 1≤i≤n, if q / 4≤z′ i If z' ≤ 3q / 4, then z' i =1; otherwise, z′ i =0, output z′ as the decrypted message.

[0021] Secondly, the present invention provides a public-key encryption system based on a novel LWE problem, comprising:

[0022] The problem definition module is configured to define novel LWE problems based on symmetric groups over n-ary sets.

[0023] The public key encryption module is configured to: construct a public key encryption scheme based on the novel LWE problem, wherein the public key encryption scheme includes decryption key generation, encryption message encryption by the encryptor, and decryption message decryption by the decryptor.

[0024] Thirdly, the present invention provides an electronic device including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the method described in the first aspect.

[0025] Fourthly, the present invention provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the method described in the first aspect.

[0026] Fifthly, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the method described in the first aspect.

[0027] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0028] This disclosure proposes a public-key encryption method and system based on a novel LWE problem. Using a symmetric group over an n-ary set as the underlying structure, a novel group-ring LWE problem is proposed, providing more candidate difficult problems for constructing quantum-resistant cryptographic schemes. Based on this, a public-key encryption scheme based on this LWE problem is proposed. This scheme can serve as a candidate quantum-resistant public-key encryption scheme for protecting the confidentiality of important data.

[0029] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0030] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0031] Figure 1 The flowchart shows the public key encryption method based on the novel LWE problem provided in Embodiment 1 of the present invention. Detailed Implementation

[0032] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0033] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0034] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments of the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. Furthermore, it should be understood that the terms “comprising” and “having”, and any variations thereof, are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0035] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0036] Example 1

[0037] like Figure 1 As shown, this embodiment provides a public-key encryption method based on a novel LWE problem, including the following steps:

[0038] S1 defines a new LWE problem based on the symmetric group over an n-ary set;

[0039] S2 constructs a public-key encryption scheme based on a novel LWE problem. The public-key encryption scheme includes key generation by the decryptor, encryption of messages by the encryptor, and decryption of messages by the decryptor.

[0040] This disclosure provides a public-key encryption method based on a novel LWE problem, which is based on the group ring LWE problem of symmetric groups over n-ary sets, and specifically includes an encryptor and a decryptor.

[0041] First, we describe the proposed novel LWE problem and its definition.

[0042] Group rings: Given a finite group G = {g1, ..., g} n The elements in the commutative ring R[G] (where n is the order of the group) are of the form and

[0043] The symmetric group S on an n-ary set n For a finite set Σ={a1,…,a…} n}, the symmetric group S defined on the n-ary set Σ n The elements can be represented as permutations The product of two permutations is defined as στ(a i )=σ(τ(a i ), where 1≤i≤n, and the inverse of permutation σ is A new type of LWE problem is defined based on the symmetric group over an n-ary set, specifically:

[0044] Given a symmetric group S on an n-ary set n Construct group ring R q =F q [S n ], where q is the modulus, F q It is a finite field modulo q, given a vector Let β be a real number and y be a vector y = xs + t, where m represents the dimension of the vector and s ← U(R). q ), U(·) is a uniform distribution. express The distribution has a Gaussian upper deviation of β. An example of the search problem is to find the secret s given x and y, based on symmetric groups and constructing group rings; an example of the decision problem is to find the secret s given x and y, and the combination of (x, y = xs + t) and a sequence from R. q ×R q Distinguish (x,y) from the uniform distribution above.

[0045] The group ring is constructed based on a finite group and a commutative ring, and the symmetric group is the set of all permutations on an n-ary set.

[0046] The proposed novel LWE problem enables the construction of feature-rich cryptographic applications. This disclosure provides a public-key encryption scheme based on this novel LWE problem, specifically including the following steps:

[0047] Step 1: Decryption Key Generation. The decryptor selects a uniformly random element a∈R. q And two random small elements s∈R, in, R represents q A Gaussian distribution with upper variance β is used. The parameter b = as + e is calculated. The output s is used as the private key. As a public key.

[0048] Step 2: The encryptor encrypts the message. To encrypt an n! bit message z∈R2, where R2 represents the group ring R2=F2[S n The encryptor selects three random elements. Calculate the parameter u = r·a + e1modq and the parameter Among them, symbols This represents rounding down the elements, where a and b are the public keys of the decryptor. The output (u, v) is used as the ciphertext of z.

[0049] Step 3: The decryptor decrypts. Calculation For 1≤i≤n, if q / 4≤z′ i If z' ≤ 3q / 4, then z' i =1; otherwise, z′ i=0. Output z′. For appropriate parameter selection, the coefficient of r·e-e1·s+e2 is less than q / 4.

[0050] Example 2

[0051] This embodiment provides a public-key encryption system based on a novel LWE problem, including:

[0052] The problem definition module is configured to define novel LWE problems based on symmetric groups over n-ary sets.

[0053] The public key encryption module is configured to: construct a public key encryption scheme based on the novel LWE problem, wherein the public key encryption scheme includes decryption key generation, encryption message encryption by the encryptor, and decryption message decryption by the decryptor.

[0054] It should be noted that the above modules correspond to the steps described in Embodiment 1, and the examples and application scenarios implemented by the above modules and the corresponding steps are the same, but are not limited to the content disclosed in Embodiment 1. It should also be noted that the above modules, as part of the system, can be executed in a computer system such as a set of computer-executable instructions.

[0055] In further embodiments, the following is also provided:

[0056] An electronic device includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the method described in Embodiment 1. For brevity, further details are omitted here.

[0057] It should be understood that in this embodiment, the processor can be a central processing unit (CPU), or it can be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor can be a microprocessor or any conventional processor, etc.

[0058] Memory may include read-only memory and random access memory, and provides instructions and data to the processor. A portion of memory may also include non-volatile random access memory. For example, memory may also store information about the device type.

[0059] A computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the method described in Embodiment 1.

[0060] The method in Example 1 can be directly implemented by a hardware processor, or implemented by a combination of hardware and software modules within the processor. The software modules can reside in readily available storage media in the field, such as random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, or registers. This storage medium is located in memory, and the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above method. To avoid repetition, a detailed description is not provided here.

[0061] A computer program product includes a computer program that, when executed by a processor, implements the method described in Embodiment 1.

[0062] The present invention also provides at least one computer program product tangibly stored on a non-transitory computer-readable storage medium. The computer program product includes computer-executable instructions, such as instructions included in program modules, which execute in a device on a target real or virtual processor to perform the processes / methods described above. Typically, program modules include routines, programs, libraries, objects, classes, components, data structures, etc., that perform specific tasks or implement specific abstract data types. In various embodiments, the functionality of program modules can be combined or divided among program modules as needed. The machine-executable instructions for the program modules can execute within a local or distributed device. In a distributed device, the program modules can reside in both local and remote storage media.

[0063] The computer program code used to implement the methods of the present invention may be written in one or more programming languages. This computer program code may be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing device, such that when executed by the computer or other programmable data processing device, the program code causes the functions / operations specified in the flowcharts and / or block diagrams to be implemented. The program code may be executed entirely on a computer, partially on a computer, as a stand-alone software package, partially on a computer and partially on a remote computer, or entirely on a remote computer or server.

[0064] In the context of this invention, computer program code or related data may be carried by any suitable carrier to enable a device, apparatus, or processor to perform the various processes and operations described above. Examples of carriers include signals, computer-readable media, and so on. Examples of signals may include electrical, optical, radio, sound, or other forms of propagation signals, such as carrier waves, infrared signals, etc.

[0065] Those skilled in the art will recognize that the units and algorithm steps described in conjunction with the embodiments herein can be implemented in electronic hardware or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0066] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A public key encryption method based on a new LWE problem, characterized by, The method comprises the following steps: Based on Symmetric group on the set of elements defines a new type of LWE problem; Specifically: Given Symmetric group on the primal set Constructing group rings ,in, q It's the modulus. It is a model q Finite field, given vector real numbers , and vector ,in, This represents the dimension of a vector. , , To ensure uniform distribution, express The difference above is The Gaussian distribution of , based on the symmetry group and the construction of group rings, given and Seeking the secret The group ring is constructed based on finite groups and commutative rings, and the symmetric group is... The set consisting of all permutations on the metaset; The public key encryption scheme is constructed based on the new LWE problem, and the public key encryption scheme comprises decryption party key generation, encryption party message encryption, and decryption party message decryption. The decrypting party chooses a uniformly random element and two random small elements R, where denotes a Gaussian distribution with variance and outputs as the private key, as the public key; The encryption party message encryption comprises the following steps: The encryptor selects three random elements , calculates the parameter and the parameter , and outputs as the ciphertext of the message to be encrypted; The decryption party message decryption comprises the following steps: For if then ; otherwise, output is the decrypted message.

2. A public key encryption system based on a new LWE problem, characterized by, The method comprises the following steps: a problem definition module configured to define a problem based on a symmetric group over a set of elements defines a new type of LWE problem; The method comprises the following steps:

3. An electronic device, comprising: The public key encryption module is configured to construct a public key encryption scheme based on the new LWE problem, and the public key encryption scheme comprises decryption party key generation, encryption party message encryption, and decryption party message decryption.

4. A computer-readable storage medium, characterized in that, The computer program product comprises a memory and a processor, and computer instructions stored in the memory and run on the processor, and when the computer instructions are run by the processor, the method of claim 1 is completed.

5. A computer program product, characterised in that, The computer program product comprises a memory and a processor, and computer instructions stored in the memory and run on the processor, and when the computer instructions are run by the processor, the method of claim 1 is completed. The computer program product comprises a memory and a processor, and computer instructions stored in the memory and run on the processor, and when the computer instructions are run by the processor, the method of claim 1 is completed.

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